The final part of a question presents the "Backwards Euler method", as:
x(n+1) = x(n) + h * f( (n+1) * h, x(n+1) ) (where f(t,x) = dx/dt)
It then says to apply this method to the linear equation dx/dt=x, and show that the method converges to the true solution x(t)=e^t as t->infinity.
It is obvious that the term (1+t/n)^n will turn up at some stage (seeing as the limit of it, as t->infinity, is e^t). But I'm struggling to get started. My problem is that in order to find the derivative at x(n+1), I must find what x(n+1) is; but to find x(n+1), I need to know the derivative at x(n+1)...
So, can someone at least get me past this hurdle, then I will hopefully be able to figure the rest for myself.
An earlier part of this question might be relevent; it says to apply Euler's method to dx/dt=kx. For which I get: